Quantum -potentials associated to quantum Ornstein–Uhlenbeck semigroups

Abstract

Using the quantum Ornstein–Uhlenbeck (O–U) semigroups (introduced in Rguigui [21]) and based on nuclear infinite dimensional algebra of entire functions with a certain exponential growth condition with two variables, the quantum -potential and the generalised quantum -potential appear naturally for . We prove that the solution of Poisson equations associated with the suitable quantum number operators can be expressed in terms of these potentials. Using a useful criterion for the positivity of generalised operators, we demonstrate that the solutions of the Cauchy problems associated to the quantum number operators are positive operators if the initial condition is also positive. In this case, we show that these solutions, the quantum -potential and the generalised quantum -potential have integral representations given by positive Borel measures. Based on a new notion of positivity of white noise operators, the aforementioned potentials are shown to be Markovian operators whenever and .

Full text

Quantum k-potentials associated to quantum
Ornstein–Uhlenbeck semigroups
Hafedh Rguigui
Department of Mathematics, High School of Sciences and Technology of Hammam Sousse, University of Sousse, Rue Lamine Abassi, 4011 Hammam Sousse, Tunisia
article info
Article history:
Received 17 October 2014
Accepted 3 January 2015
abstract
Using the quantum Ornstein–Uhlenbeck (O–U) semigroups (introduced in Rguigui [21])
and based on nuclear infinite dimensional algebra of entire functions with a certain expo-
nential growth condition with two variables, the quantum k-potential and the generalised
quantum ðk
1
;k
2
Þ-potential appear naturally for k;k
1
;k
2
2ð0;1Þ. We prove that the solution
of Poisson equations associated with the suitable quantum number operators can be
expressed in terms of these potentials. Using a useful criterion for the positivity of
generalised operators, we demonstrate that the solutions of the Cauchy problems associ-
ated to the quantum number operators are positive operators if the initial condition is also
positive. In this case, we show that these solutions, the quantum k-potential and the gen-
eralised quantum ðk
1
;k
2
Þ-potential have integral representations given by positive Borel
measures. Based on a new notion of positivity of white noise operators, the aforementioned
potentials are shown to be Markovian operators whenever k2½1;1Þ and k
1
k
2
Pk
1
þk
2
.
Ó2015 Elsevier Ltd. All rights reserved.
1. Introduction
The term ‘‘Potential’’ is derived from mathematical
physics, in particular from gravitational problems and elec-
trostatic, where the fundamental forces which are closely
related to the potential energy like gravity or electrostatic
forces were given as the gradients of potentials, i.e., func-
tions which satisfy the Laplace equation and more gener-
ally the Poisson equation. This last equation is still used
in applications in many physics fields and engineering like
heat conduction and electrostatics. Later on, using the
notion of Radon measure, positivity and generalised func-
tions, the generalisation of the principal problems and
the completion of the existing formulations in the poten-
tial theory were made. The fundamental dynamical charac-
terisation of the contraction property of an energy
functional f on a special Hilbert space was discovered
(see [7] and references cited therein), in terms of the semi-
group by the Hamiltonian H: the contraction property of f
is equivalent to the so called Markovianity property, which
consists of positivity, in the sense that the maps the semi-
group preserve positivity of functions, and of contractivity,
by which the semigroup is contractive both with respect to
the Hilbertian norm and with respect to the uniform norm
of a particular algebra. The modern period of the develop-
ment of potential theory is characterised by the applica-
tions of methods and notions of topology and functional
analysis. Since the Laplace equation and more generally
the Poisson equation in fintie dimensional spaces need
the Laplacian operator to be defined, then to give an infi-
nite dimensional analogue of the potential theory they
need to extend the former. Piech [20] introduced the
number operator N(Beltrami Laplacian) as infinite dimen-
sional analogue of a finite dimensional Laplacian. This infi-
nite dimensional Laplacian has been extensively studied in
[14,15] and the references cited therein. In particular, Kuo
[14] formulated the number operator as continuous linear
operator acting on the space of test white noise function-
als. Kuo [13] and Kang [12] have proved some theorems
in potential theory with respect to Ornstein–Uhlenbeck
http://dx.doi.org/10.1016/j.chaos.2015.01.001
0960-0779/Ó2015 Elsevier Ltd. All rights reserved.
E-mail address: hafedh.rguigui@yahoo.fr
Chaos, Solitons & Fractals 73 (2015) 80–89
Contents lists available at ScienceDirect
Chaos, Solitons & Fractals
Nonlinear Science, and Nonequilibrium and Complex Phenomena
journal homepage: www.elsevier.com/locate/chaos

(O–U) semigroup in abstract Wiener space, they have stud-
ied the Poisson and the heat equation associated with the
number operator N, these solutions are related to the O–
U semigroup. Kang [12] has proved that the solutions of
generalised Poisson equations associated with number
operator are represented by the k-potentials in white noise
setting. In [5], based on nuclear algebra of entire functions,
some results are extended about operator-parameter
transforms involving the O–U semigroup. In [21] the quan-
tum analogue of the O–U semigroups are given, moreover
the heat equations associated to suitable quantum number
operators are solved.
In this paper, with reference to a nuclear algebra of
entire functions with two variables, the aforementioned
quantum potentials will be defined and we prove that
the solutions of Poisson equations associated with their
suitable quantum number operators can be expressed in
terms of these potentials, see Section 3. Moreover, in
Section 4, using the definition of the positivity of
generalised white noise operators, we prove that the
solutions of the Cauchy problems associated to the quan-
tum number operators are positive operators if the initial
condition is also positive. In this case, we reveal that these
solutions and these quantum potentials have integral
representations given by positive Borel measure. In Section
5, based on a cone obtained from a new notion of positivity
of white noise operators, the quantum potentials are
shown to be Markovian operators whenever k2½1;1Þ
and k
1
k
2
Pk
1
þk
2
.
2. Preliminaries and theoretical results
First we review basic concepts, notations, and some
results which will be needed in the present paper. The
development of the later along with the results can be
found in Refs. [1,3–6,8–11,16,19].
For i¼1;2, let N
i
be the complexification of a real
nuclear Fréchet space X
i
whose topology is defined by a
family of increasing Hilbert norms fj:j
p
;p2Ng. For p2N,
we denote by ðN
i
Þ
p
the completion of N
i
with respect to
the norm j:j
p
and by ðN
i
Þ
p
respectively N
0
i
the strong dual
space of ðN
i
Þ
p
and N
i
. Then, we obtain
N
i
¼projlim
p!1
ðN
i
Þ
p
and N
0
i
¼indlim
p!1
ðN
i
Þ
p
:ð1Þ
The spaces N
i
and N
0
i
are respectively equipped with the
projective and inductive limit topology. For all p2N,we
denote by j:j
p
the norm on ðN
i
Þ
p
and by h:; :ithe Cbilin-
ear form on N
0
i
N
i
. In the following, Hdenote by the direct
Hilbertian sum of ðN
1
Þ
0
and ðN
2
Þ
0
, i.e., H¼ðN
1
Þ
0
ðN
2
Þ
0
.
For n2N, we denote by Nb
n
i
the n-fold symmetric tensor
product on N
i
equipped with the
p
topology and by
ðN
i
Þb
n
p
the n-fold symmetric Hilbertian tensor product on
ðN
i
Þ
p
. We will preserve the notation j:j
p
and j:j
p
for the
norms on ðN
i
Þb
n
p
and ðN
i
Þb
n
p
, respectively.
Throughout the paper, we fix a pair of Young function
ðh
1
;h
2
Þ. From now on we assume that the Young functions
h
i
satisfy
lim
r!1
h
i
ðrÞ
r
2
<1:ð2Þ
For all positive numbers m
1
;m
2
>0 and all integers
ðp
1
;p
2
Þ2NN, we define the space of all entire functions
on ðN
1
Þ
p
1
ðN
2
Þ
p
2
with ðh
1
;h
2
Þexponential growth by
ExpððN
1
Þ
p
1
ðN
2
Þ
p
2
;ðh
1
;h
2
Þ;ðm
1
;m
2
ÞÞ
¼f2 HððN
1
Þ
p
1
ðN
2
Þ
p
2
Þ;kfk
ðh
1
;h
2
Þ;ðp
1
;p
2
Þ;ðm
1
;m
2
Þ
<1
no
where HððN
1
Þ
p
1
ðN
2
Þ
p
2
Þis the space of all entire func-
tions on ðN
1
Þ
p
1
ðN
2
Þ
p
2
and
kfk
ðh
1
;h
2
Þ;ðp
1
;p
2
Þ;ðm
1
;m
2
Þ
¼sup jfðz
1
;z
2
Þje
h
1
ðm
1
jz
1
j
p1
Þh
2
ðm
2
jz
2
j
p2
Þ
no
for ðz
1
;z
2
Þ2ðN
1
Þ
p
1
ðN
2
Þ
p
2
. So, the space of all entire
functions on ðN
1
Þ
p
1
ðN
2
Þ
p
2
with ðh
1
;h
2
Þ-exponential
growth of minimal type is naturally defined by
F
h
1
;h
2
ðN
0
1
N
0
2
Þ¼proj lim
p
1
;p
2
!1;m
1
;m
2
#0
ExpððN
1
Þ
p
1
ðN
2
Þ
p
2
;ðh
1
;h
2
Þ;ðm
1
;m
2
ÞÞ:ð3Þ
By definition,
u
2F
h
1
;h
2
ðN
0
1
N
0
2
Þadmits the Taylor
expansions:
u
ðx;yÞ¼ X
1
n;m¼0
x
n
y
m
;
u
n;m
DE
;ðx;yÞ2N
0
1
N
0
2
ð4Þ
where for all n;m2Nwe have
u
n;m
2Nb
n
1
Nb
m
2
and we
used the common symbol h:; :ifor the canonical Cbilinear
form on ðN
n
1
N
m
2
Þ0  N
n
1
N
m
2
. So, we identify in the
next all test function
u
2F
h
1
;h
2
ðN
0
1
N
0
2
Þby their coeffi-
cients of its Taylors series expansion at the origin
ð
u
n;m
Þ
n;m2N
. As important example of elements in
F
h
1
;h
2
ðN
0
1
N
0
2
Þ, we define the exponential function as fol-
lows. For a fixed ðn;
g
Þ2N
1
N
2
,
e
ðn;
g
Þ
ða;bÞ¼ðe
n
e
g
Þða;bÞ¼expfha;niþhb;
g
ig;
ða;bÞ2N
0
1
N
0
2
:
Denoted by F

h
1
;h
2
ðN
0
1
N
0
2
Þthe topological dual of
F
h
1
;h
2
ðN
0
1
N
0
2
Þcalled the space of distribution on
N
0
1
N
0
2
. In the particular case where N
2
¼f0g, we obtain
the following identification
F
h
1
;h
2
ðN
0
1
f0gÞ ¼ F
h
1
ðN
0
1
Þ
and therefore
F

h
1
;h
2
ðN
0
1
f0gÞ ¼ F

h
1
ðN
1
0Þ:
So, the space F
h
1
;h
2
ðN
0
1
N
0
2
Þcan be considered as a gener-
alisation of the space F
h
1
ðN
1
0Þ studied in [8].
Denoting by LðX;YÞto be the space of continuous linear
operators from a nuclear space Xto another nuclear space
Y. From the nuclearity of the spaces F
h
i
ðN
0
i
Þ, we have by
Kernel Theorem the following isomorphisms
LðF

h
1
ðN
0
1
Þ;F
h
2
ðN
0
2
ÞÞ ’ F
h
1
ðN
0
1
ÞF
h
2
ðN
0
2
Þ
’F
h
1
;h
2
ðN
0
1
N
0
2
Þ:ð5Þ
So, for every
N
2 LðF

h
1
ðN
0
1
Þ;F
h
2
ðN
0
2
ÞÞ, the associated kernel
U
N
2F
h
1
;h
2
ðN
0
1
N
0
2
Þis defined by
H. Rguigui / Chaos, Solitons & Fractals 73 (2015) 80–89 81

hh
N
u
;wii ¼ hh
U
N
;
u
wii;
8
u
2F

h
1
ðN
0
1
Þ;
8
w2F

h
2
ðN
0
2
Þ:
ð6Þ
Denoting by Kthe topological isomorphism:
LðF

h
1
ðN
0
1
Þ;F
h
2
ðN
0
2
ÞÞ 3
N
#K
N
¼
U
N
2F
h
1
;h
2
ðN
0
1
N
0
2
Þ:ð7Þ
2.1. Quantum number operator
Recall that the standard number operator on F
h
ðN0Þ is
given by
N
u
ðxÞ¼X
1
n¼1
hx
n
;n
u
n
i;ð8Þ
where
u
ðxÞ¼P
1
n¼0
hx
n
;
u
n
i2F
h
ðN0Þ, see [14] and refer-
ences cited therein. For
u
ðx;yÞ¼P
1
n;m¼0
hx
n
y
m
;
u
n;m
i
in F
h
1
;h
2
ðN
0
1
N
0
2
Þ, the three following operators (see [21])
are defined by:
N
u
ðx;yÞ:¼X
1
n;m¼0;ðn;mÞ–ð0;0Þ
ðnþmÞx
n
y
m
;
u
n;m
DE
:ð9Þ
N
1
u
ðx;yÞ:¼X
1
n¼1;m¼0
nhx
n
y
m
;
u
n;m
i;ð10Þ
N
2
u
ðx;yÞ:¼X
1
n¼0;m¼1
mhx
n
y
m
;
u
n;m
i:ð11Þ
It was shown that N;N
1
and N
2
are linear continuous
operators from F
h
1
;h
2
ðN
0
1
N
0
2
Þinto itself. Moreover, from
(9) and (8) we can easily see that N
1
;N
2
and Nhave the
following decompositions
N
1
¼NI;N
2
¼IN;N¼NIþIN;
respectively. As quantum analogues of the number opera-
tor, the following operators were introduced, see [21].
Definition 2.1. We define the following operator on
LðF

h
1
ðN
0
1
Þ;F
h
2
ðN
0
2
ÞÞ by
f
N
1
:¼K
1
ðN
1
ÞK;f
N
2
:¼K
1
ðN
2
ÞK;f
N:¼K
1
NK
¼f
N
1
þf
N
2
:
The operator f
N
1
is called left quantum number operator,
f
N
2
is called right quantum number operator and f
Nis
called quantum number operator.
Note that Definition 2.1 holds true on LðF
h
1
ðN
0
1
Þ;
F

h
2
ðN
0
2
ÞÞ.
2.2. Cauchy problem associated with quantum number
operator
In [21], it was constructed a semigroup e
Q
t
;tP0
no
;
e
Q
s;0
;sP0
no
and fe
Q
0;t
;tP0gon LðF

h
1
ðN
0
1
Þ;F
h
2
ðN
0
2
ÞÞ
with infinitesimal generator f
N;f
N
1
and f
N
2
, respec-
tively. It reminds to construct a semigroup fQ
t
;tP0g;
fQ
s;0
;sP0gand fQ
0;t
;tP0gon F
h
1
;h
2
ðN
0
1
N
0
2
Þwith
infinitesimal generator N ;N
1
and N
2
, respectively.
Observe that symbolically Q
s;t
¼e
sN
1
tN
2
. Thus, we can
define the operator Q
s;t
as follows. For
u
ð
u
n;m
Þ,wedefine
Q
s;t
u
ðx;yÞ:¼X
1
n;m¼0
x
n
y
m
;e
sntm
u
n;m
DE
;ð12Þ
and let Q
t;t
denoted by Q
t
. It was shown (see [21]) that, for
any s;tP0, the linear operator Q
s;t
is continuous from
F
h
1
;h
2
ðN
0
1
N
0
2
Þinto itself. More precisely, for m
0
1
;m
0
2
>
0;m
1
;m
2
>0;q
1
>p
1
and q
2
>p
2
such that
max m
1
m
0
1
eki
q
1
;p
1
k
HS
;m
2
m
0
2
eki
q
2
;p
2
k
HS

<1;
we get
kQ
s;t
u
k
ðh
1
;h
2
Þ;ðp
1
;p
2
Þ;ðm
0
1
;m
0
2
Þ
6k
u
k
ðh
1
;h
2
Þ;ðq
1
;q
2
Þ;ðm
1
;m
2
Þ
K
p
1
;p
2
;q
1
;q
2
ð13Þ
where K
p
1
;p
2
;q
1
;q
2
is given by
K
p
1
;p
2
;q
1
;q
2
¼1m
1
m
0
1
eki
q
1
;p
1
k
HS

1
1m
2
m
0
2
eki
q
2
;p
2
k
HS

1
:
It was shown that
e
Q
s;t
:¼K
1
Q
s;t
K¼e
O
s;t
2 LðLðF

h
1
ðN
0
1
Þ;F
h
2
ðN
0
2
ÞÞÞ;
where g
O
s;t
¼K
1
O
s;t
K,
O
s;t
u
ðy
1
;y
2
Þ¼Z
X
0
1
X
0
2
u
ðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1expð2sÞ
px
1
þexpðsÞy
1
;ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1expð2tÞ
px
2
þexpðtÞy
2
Þd
l
1
ðx
1
Þd
l
2
ðx
2
Þ
and
l
j
is the standard Gaussian measure on X
j
0ðforj ¼1;2Þ
uniquely specified by its characteristic function
e

1
2
jnj
2
0
¼Z
X
j
0
e
ihx;ni
l
j
ðdxÞ;n2X
j
:
The operator e
Q
t;t
¼g
O
t;t
, denoted by f
Q
t
for simplicity, is
called the quantum O–U semigroup. The operator
e
Q
s;0
¼g
O
s;0
is called the left quantum O–U semigroup and
the operator e
Q
0;t
¼g
O
0;t
is called the right quantum O–U
semigroup.
Theorem 2.1 [21].The families fe
Q
t
;tP0g;fe
Q
s;0
;sP0g
and fe
Q
0;t
;tP0gare strongly continuous semigroup of
continuous linear operators from LðF

h
1
ðN
0
1
Þ;F
h
2
ðN
0
2
ÞÞ into
itself with the infinitesimal generator f
N;g
N
1
and g
N
2
respectively. Moreover, the quantum Cauchy problems
dP
t
dt
¼
f
N
P
t
P
0
¼
N
2 LðF

h
1
ðN
0
1
Þ;F
h
2
ðN
0
2
ÞÞ
(ð14Þ
dK
s
ds
¼
g
N
1
K
s
K
0
¼
N
2 LðF

h
1
ðN
0
1
Þ;F
h
2
ðN
0
2
ÞÞ
(ð15Þ
82 H. Rguigui / Chaos, Solitons & Fractals 73 (2015) 80–89

d
!
t
dt
¼
g
N
2
!
t
!
0
¼
N
2 LðF

h
1
ðN
0
1
Þ;F
h
2
ðN
0
2
ÞÞ
(ð16Þ
have a unique solutions given respectively by
P
t
¼e
Q
t
N
;
K
s
¼e
Q
s;0
N
and !
t
¼e
Q
0;t
N
:ð17Þ
3. Quantum k-potentials associated to quantum
Ornstein–Uhlenbeck semigroups
For k;k
1
;k
2
>0 and
u
on F
h
1
;h
2
ðN
0
1
N
0
2
Þ, we define the
following operators:
H
k
u
ðx;yÞ¼Z
1
0
e
kt
Q
t
u
ðx;yÞdt ð18Þ
H
g
k
1
;k
2
u
ðx;yÞ¼H

k
1
u
ðx;yÞþH
þ
k
2
u
ðx;yÞ;ð19Þ
where H

k
1
and H
þ
k
2
are given by
H

k
1
u
ðx;yÞ¼Z
1
0
e
k
1
s
Q
s;0
u
ðx;yÞds ð20Þ
H
þ
k
2
u
ðx;yÞ¼Z
1
0
e
k
2
t
Q
0;t
u
ðx;yÞdt:ð21Þ
In view of (13),e
kt
Q
t
u
;e
k
1
s
Q
s;0
u
and e
k
2
t
Q
0;t
u
are inte-
grable and bounded by
e
kt
k
u
k
ðh
1
;h
2
Þ;ðq
1
;q
2
Þ;ðm
1
;m
2
Þ
e
h
1
ðm
0
1
jxj
p1
Þþh
2
ðm
0
2
jyj
p2
Þ
K
p
1
;p
2
;q
1
;q
2
e
k
1
s
k
u
k
ðh
1
;h
2
Þ;ðq
1
;q
2
Þ;ðm
1
;m
2
Þ
e
h
1
ðm
0
1
jxj
p1
Þþh
2
ðm
0
2
jyj
p2
Þ
K
p
1
;p
2
;q
1
;q
2
e
k
2
t
k
u
k
ðh
1
;h
2
Þ;ðq
1
;q
2
Þ;ðm
1
;m
2
Þ
e
h
1
ðm
0
1
jxj
p1
Þþh
2
ðm
0
2
jyj
p2
Þ
K
p
1
;p
2
;q
1
;q
2
respectively. Remark that the integrand in the right hand
side of (18), (20) and (21) are not integrable when
k!0;k
1
!0 and k
2
!0, respectively. For this, define the
following normalised operators:
ðG
u
Þðx;yÞ¼Z
1
0
Q
t
ð
u

u
0;0
Þðx;yÞdt;ð22Þ
ðG
g
u
Þðx;yÞ¼ðG

u
Þðx;yÞþðG
þ
u
Þðx;yÞð23Þ
where
u
ðx;yÞ¼P
1
n;m¼0
hx
n
y
m
;
u
n;m
i;
u

0
ðx;yÞ¼ P
1
m¼1
hy
m
;
u
0;m
i;
u
þ
0
ðx;yÞ¼P
1
n¼1
hx
n
;
u
n;0
iand
ðG

u
Þðx;yÞ¼Z
1
0
Q
s;0
ð
u

u
0;0

u

0
Þðx;yÞds;ð24Þ
ðG
þ
u
Þðx;yÞ¼Z
1
0
Q
0;t
ð
u

u
0;0

u
þ
0
Þðx;yÞdt:ð25Þ
Using (12), we observe that
Q
t
ð
u

u
0;0
Þðx;yÞ¼ X
1
n;m¼0;ðn;mÞ–ð0;0Þ
x
n
y
m
;e
tðnþmÞ
u
n;m
DE
Q
s;0
ð
u

u
0;0

u

0
Þðx;yÞ¼ X
1
n¼1;m¼0
x
n
y
m
;e
sn
u
n;m
DE
Q
0;t
ð
u

u
0;0

u
þ
0
Þðx;yÞ¼ X
1
n¼0;m¼1
x
n
y
m
;e
tm
u
n;m
DE
and for lP1 and rP0, we have e
rl
6e
r
. Then,
Q
t
ð
u

u
0;0
Þðx;yÞ;Q
s;0
ð
u

u
0;0

u

0
Þðx;yÞand Q
0;t
ð
u

u
0;0

u
þ
0
Þðx;yÞare bounded by the integrable functions
e
t
k
u
k
ðh
1
;h
2
Þ;ðq
1
;q
2
Þ;ðm
1
;m
2
Þ
e
h
1
ðm
0
1
jxj
p1
Þþh
2
ðm
0
2
jyj
p2
Þ
K
p
1
;p
2
;q
1
;q
2
e
s
k
u
k
ðh
1
;h
2
Þ;ðq
1
;q
2
Þ;ðm
1
;m
2
Þ
e
h
1
ðm
0
1
jxj
p1
Þþh
2
ðm
0
2
jyj
p2
Þ
K
p
1
;p
2
;q
1
;q
2
e
t
k
u
k
ðh
1
;h
2
Þ;ðq
1
;q
2
Þ;ðm
1
;m
2
Þ
e
h
1
ðm
0
1
jxj
p1
Þþh
2
ðm
0
2
jyj
p2
Þ
K
p
1
;p
2
;q
1
;q
2
respectively.
Lemma 3.1. H
k
;H
g
k
1
;k
2
;Gand G
g
are continuous linear map-
pings from F
h
1
;h
2
ðN
0
1
N
0
2
Þ. Moreover, for q
1
>p
1
;q
2
>p
2
and m
1
;m
2
,m
0
1
;m
0
2
>0such that
max m
1
m
0
1
eki
q
1
;p
1
k
HS
;m
2
m
0
2
eki
q
2
;p
2
k
HS

<1;
we have, for any
u
2F
h
1
;h
2
ðN
0
1
N
0
2
Þ,
H
k
u
kk
ðh
1
;h
2
Þ;ðp
1
;p
2
Þ;ðm
0
1
;m
0
2
Þ
61
kk
u
k
ðh
1
;h
2
Þ;ðq
1
;q
2
Þ;ðm
1
;m
2
Þ
K
p
1
;p
2
;q
1
;q
2
Hg
k
1
;k
2
u




ðh
1
;h
2
Þ;ðp
1
;p
2
Þ;ðm
0
1
;m
0
2
Þ
61
k1
þ1
k2

k
u
kðh
1
;h
2
Þ;ðq
1
;q
2
Þ;ðm
1
;m
2
ÞKp
1
;p
2
;q
1
;q
2
and
G
u
kk
ðh
1
;h
2
Þ;ðp
1
;p
2
Þ;ðm
0
1
;m
0
2
Þ
6k
u
k
ðh
1
;h
2
Þ;ðq
1
;q
2
Þ;ðm
1
;m
2
Þ
K
p
1
;p
2
;q
1
;q
2
G
g
u



ðh
1
;h
2
Þ;ðp
1
;p
2
Þ;ðm
0
1
;m
0
2
Þ
62k
u
k
ðh
1
;h
2
Þ;ðq
1
;q
2
Þ;ðm
1
;m
2
Þ
K
p
1
;p
2
;q
1
;q
2
:
Proof. Recall that e
kt
Q
t
u
ðx;yÞis bounded by
e
kt
k
u
k
ðh
1
;h
2
Þ;ðq
1
;q
2
Þ;ðm
1
;m
2
Þ
e
h
1
ðm
0
1
jxj
p1
Þþh
2
ðm
0
2
jyj
p2
Þ
K
p
1
;p
2
;q
1
;q
2
and Q
t
is a continuous linear map. By the Lebesgue domi-
nated convergence theorem, we obtain
H
k
u
ðx;yÞ¼ X
1
n;m¼0
Z
1
0
e
kt
x
n
y
m
;e
tðnþmÞ
u
n;m
DE
dt
¼X
1
n;m¼0
x
n
y
m
;
u
n;m
DE
Z
1
0
e
tðkþnþmÞ
dt
¼X
1
n;m¼0
1
kþnþmx
n
y
m
;
u
n;m
DE
:
Which is equivalent to write
H
k
u
1
kþnþm
u
n;m

:ð26Þ
Similarly, we get
H

k
1
u
1
k
1
þn
u
n;m
 ð27Þ
H
þ
k
2
u
1
k
2
þm
u
n;m

:ð28Þ
Then, we obtain
H
g
k
1
;k
2
u
1
k
1
þnþ1
k
2
þm

u
n;m

:ð29Þ
H. Rguigui / Chaos, Solitons & Fractals 73 (2015) 80–89 83

For any p
1
;p
2
P0, we get
jH
k
u
ðx;yÞj 6X
1
n;m¼0
1
kþnþmjxj
n
p
1
jyj
m
p
2
j
u
n;m
j
p
1
;p
2
61
kX
1
n;m¼0
jxj
n
p
1
jyj
m
p
2
j
u
n;m
j
p
1
;p
2
:
Then, as in the proof of (13) (see [21]), we get
H
k
u
kk
ðh
1
;h
2
Þ;ðp
1
;p
2
Þ;ðm
0
1
;m
0
2
Þ
61
kk
u
k
ðh
1
;h
2
Þ;ðq
1
;q
2
Þ;ðm
1
;m
2
Þ
K
p
1
;p
2
;q
1
;q
2
:
Similarly, we obtain
H

k
1
u





ðh
1
;h
2
Þ;ðp
1
;p
2
Þ;ðm
0
1
;m
0
2
Þ
61
k
1
k
u
k
ðh
1
;h
2
Þ;ðq
1
;q
2
Þ;ðm
1
;m
2
Þ
K
p
1
;p
2
;q
1
;q
2
H
þ
k
2
u





ðh
1
;h
2
Þ;ðp
1
;p
2
Þ;ðm
0
1
;m
0
2
Þ
61
k
2
k
u
k
ðh
1
;h
2
Þ;ðq
1
;q
2
Þ;ðm
1
;m
2
Þ
K
p
1
;p
2
;q
1
;q
2
Hg
k
1
;k
2
u




ðh
1
;h
2
Þ;ðp
1
;p
2
Þ;ðm
0
1
;m
0
2
Þ
61
k1
þ1
k2

k
u
kðh
1
;h
2
Þ;ðq
1
;q
2
Þ;ðm
1
;m
2
ÞKp
1
;p
2
;q
1
;q
2
:
As above, one can show that
ðG
u
Þðx;yÞ¼ X
1
n;m¼0;ðn;mÞ–ð0;0Þ
1
nþmhx
n
y
m
;
u
n;m
ið30Þ
ðG

u
Þðx;yÞ¼ X
1
n¼1;m¼0
1
nhx
n
y
m
;
u
n;m
ið31Þ
ðG
þ
u
Þðx;yÞ¼ X
1
n¼0;m¼1
1
mhx
n
y
m
;
u
n;m
ið32Þ
Then, for any p
1
;p
2
P0
jðG
u
Þðx;yÞj 6X
1
n;m¼0;ðn;mÞ–ð0;0Þ
1
nþmjxj
n
p
1
jyj
m
p
2
j
u
n;m
j
p
1
;p
2
6X
1
n;m¼0
jxj
n
p
1
jyj
m
p
2
j
u
n;m
j
p
1
;p
2
jðG
g
u
Þðx;yÞj 6jðG

u
Þðx;yÞj þ jðG
þ
u
Þðx;yÞj
62X
1
n;m¼0
jxj
n
p
1
jyj
m
p
2
j
u
n;m
j
p
1
;p
2
Hence, similarly to (13), we complete the proof. h
For k;k
1
;k
2
>0, the quantum k-potential, the general-
ised quantum ðk
1
;k
2
Þ-potential, the quantum normalised
potential and the generalised quantum normalised poten-
tial are defined respectively on LðF

h
1
ðN
0
1
Þ;F
h
2
ðN
0
2
ÞÞ by
e
H
k
:¼K
1
H
k
K;
f
H
g
k
1
;k
2
:¼K
1
H
g
k
1
;k
2
K¼ f
H
k
1
þf
H
þk
2
¼K
1
H

k
1
KþK
1
H
þ
k
2
K;
e
G:¼K
1
GK;
f
G
g
:¼K
1
G
g
K¼ f
G

þf
G
þ
¼K
1
G

KþK
1
G
þ
K:
3.1. Poisson equation associated with quantum number
operator
Theorem 3.1. Let
N
2 LðF

h
1
ðN
0
1
Þ;F
h
2
ðN
0
2
ÞÞ, such that
Kð
N
Þ¼
u
ð
u
n;m
Þ
n;m
. Then, we have
f
Ne
G
N
¼
N

N
0;0
f
Ne
H
k
N
¼
N
ke
H
k
N
g
N
1
f
G

N
¼
N

N
0;0

N

0
g
N
1
f
H
k
1
N
¼
N
k
1
f
H
k
1
N
g
N
2
f
G
þ
N
¼
N

N
0;0

N
þ
0
g
N
2
f
H
þ
k
2
N
¼
N
k
2
f
H
þ
k
2
N
where
N
0;0
¼
u
0;0
I;
N

0
¼K
1
ð
u

0
Þand
N
þ
0
¼K
1
ð
u
þ
0
Þ.
Proof. Let
u
ð
u
n;m
Þ
n;m
in F
h
1
;h
2
ðN
0
1
N
0
2
Þ. Then, using (9)
and (30), one can obtain
NG
u
ðx;yÞ¼ X
1
n;m¼0;ðn;mÞ–ð0;0Þ
hx
n
y
m
;
u
n;m
i
¼
u
ðx;yÞ
u
0;0
:
Using (10) and (31), we get
N
1
G

u
ðx;yÞ¼ X
1
n¼1;m¼0
hx
n
y
m
;
u
n;m
i
¼
u
ðx;yÞ
u

0
ðx;yÞ
u
0;0
:
Similarly, we have
N
2
G
þ
u
ðx;yÞ¼ X
1
n¼0;m¼1
x
n
y
m
;
u
n;m
DE
¼
u
ðx;yÞ
u
þ
0
ðx;yÞ
u
0;0
:
Using (26), we get
NH
k
u
ðx;yÞ¼N X
1
n;m¼0
1
kþnþmx
n
y
m
;
u
n;m
DE
!
¼X
1
n;m¼0
nþm
kþnþmx
n
y
m
;
u
n;m
DE
¼X
1
n;m¼0
x
n
y
m
;1k
kþnþm

u
n;m

¼
u
ðx;yÞkH
k
u
ðx;yÞ:
Similarly, using (27) and (28), one can show that
N
1
H

k
1
u
ðx;yÞ¼
u
ðx;yÞk
1
H

k
1
u
ðx;yÞ:
and
N
2
H
þ
k
2
u
ðx;yÞ¼
u
ðx;yÞk
2
H
þ
k
2
u
ðx;yÞ:
Hence, using the topological isomorphism K, we complete
the proof. h
Remark 1. Let
N
2 LðF

h
1
ðN
0
1
Þ;F
h
2
ðN
0
2
ÞÞ, such that Kð
N
Þ¼
u
ð
u
n;m
Þ
n;m
. Then, by Theorem 3.1, the operators
e
H
k
N
;f
H
k
1
N
and f
H
þk
2
N
are solutions of the following Pois-
son equations
kIþf
N

V¼
N
;k
1
Iþg
N
1

V¼
N
and k
2
Iþg
N
2

V¼
N
84 H. Rguigui / Chaos, Solitons & Fractals 73 (2015) 80–89

respectively. The operators e
G
N
;e
G

N
and e
G
þ
N
are solutions
of the following Poisson equations
f
NV¼
N

N
0;0
;g
N
1
V¼
N

N
0;0

N

0
and
g
N
2
V¼
N

N
0;0

N
þ
0
respectively, where
N
0;0
¼
u
0;0
I;
N

0
¼K
1
ð
u

0
Þand
N
þ
0
¼
K
1
ð
u
þ
0
Þ.
Remark 2. Using the topological isomorphisms via the
mapping K
LðF
h
1
ðN
0
1
Þ;F

h
2
ðN
0
2
ÞÞ ’ F

h
1
;h
2
ðN
0
1
N
0
2
Þ
LðF

h
1
ðN
0
1
Þ;F
h
2
ðN
0
2
ÞÞ ’ F
h
1
;h
2
ðN
0
1
N
0
2
Þ;
we define the duality between LðF
h
1
ðN
0
1
Þ;F

h
2
ðN
0
2
ÞÞ and
LðF

h
1
ðN
0
1
Þ;F
h
2
ðN
0
2
ÞÞ by
hhhS;
N
iii :¼ hhKðSÞ;Kð
N
Þii;ð33Þ
for S2 LðF
h
1
ðN
0
1
Þ;F

h
2
ðN
0
2
ÞÞ and
N
2 LðF

h
1
ðN
0
1
Þ;F
h
2
ðN
0
2
ÞÞ.
Working on the space LðF
h
1
ðN
0
1
Þ;F

h
2
ðN
0
2
ÞÞ, we observe that
the adjoint maps f
N

;g
N
1
;g
N
2
;e
G

;e
G

;e
G
þ
;e
G
g
;e
H

k
;e
H

k
1
;
e
H
þ
k
2
and e
H
g
k
1
;k
2
(with respect to the duality (33)) are contin-
uous linear from LðF
h
1
ðN
0
1
Þ;F

h
2
ðN
0
2
ÞÞ and coincide with the
extensions of the operators f
N;g
N
1
;g
N
2
;e
G;e
G;e
G
þ
;e
G
g
;e
H
k
;
e
H

k
1
;e
H
þ
k
2
and e
H
g
k
1
;k
2
, respectively, to LðF
h
1
ðN
0
1
Þ;F

h
2
ðN
0
2
ÞÞ
(we have used the same notation for these extensions).
Moreover, Theorem 3.1 and Remark 1 hold true on
LðF
h
1
ðN
0
1
Þ;F

h
2
ðN
0
2
ÞÞ.
3.2. Quantum–classical correspondence
Let
u
ðyÞ¼P
1
n¼0
y
n
;
u
n
hi
2F
h
i
ðN
0
i
Þ;i¼1;2, the opera-
tor q
t
is defined by
q
t
u
ðyÞ:¼X
1
n¼0
y
n
;e
tn
u
n

;ð34Þ
for all tP0. q
t
can be extended to F

h
i
ðN
0
i
Þ(and denoted by
the same notation) as follows
q
t
U
ðe
tn
U
n
Þ
nP0
;
8U
ð
U
n
Þ
nP0
:
Then, q
t
2 LðF
h
1
ðN
0
1
Þ;F
h
2
ðN
0
2
ÞÞTLðF

h
1
ðN
0
1
Þ;F

h
2
ðN
0
2
ÞÞ. More-
over, it is represented by
q
t
u
ðyÞ¼Z
X
0
i
u
ðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1expð2tÞ
pxþexpðtÞyÞd
l
i
ðxÞ
on F
h
i
ðN
0
i
Þ, for more details see [14]. Using the duality
between F

h
i
ðN
0
i
Þand F
h
i
ðN
0
i
Þ, one can easily show that
q

t
¼q
t
;
8
tP0:ð35Þ
The classical k-potential (see [13]) is defined by
h
k
u
¼Z
1
0
e
kt
q
t
u
dt:
Using (35), we get
h

k
¼h
k
;
8
k2ð0;1Þ:ð36Þ
Theorem 3.2. Let k
1
;k
2
2ð0;1Þ, then
e
H
g
k
1
;k
2
ð
N
Þ¼
N
h
k
1
þh
k
2
N
for all
N
2 LðF
h
1
ðN
0
1
Þ;F
h
2
ðN
0
2
ÞÞ TLðF

h
1
ðN
0
1
Þ;F

h
2
ðN
0
2
ÞÞ or
equivalently
H
g
k
1
;k
2
¼h
k
1
IþIh
k
2
:
Proof. Applying the kernel map to e
H
g
k
1
;k
2
, we get
H
g
k
1
;k
2
¼Z
1
0
e
k
1
s
Q
s;0
ds þZ
1
0
e
k
2
t
Q
0;t
dt:
But using the fact that (see [21])
Q
s;t
¼q
s
q
t
;
8
s;t2ð0;1Þ
we obtain
H
g
k
1
;k
2
¼Z
1
0
e
k
1
s
q
s
q
0
ds þZ
1
0
e
k
2
t
q
0
q
t
dt:
Since q
0
¼I, we get
H
g
k
1
;k
2
¼h
k
1
IþIh
k
2
from which we deduce that
hh e
H
g
k
1
;k
2
ð
N
Þ
u
;wii ¼ hhH
g
k
1
;k
2
Kð
N
Þ;
u
wii
¼ hhKð
N
Þ;ðH
g
k
1
;k
2
Þ

u
wii
¼ hhKð
N
Þ;ðh
k
1
IþIh
k
2
Þ

u
wii:
Then, using (36), we get
hh e
H
g
k
1
;k
2
ð
N
Þ
u
;wii ¼ hhKð
N
Þ;ðh
k
1
IþIh
k
2
Þ
u
wii
¼ hhKð
N
Þ;h
k
1
ð
u
Þwii þ hhKð
N
Þ;
u
h
k
2
ðwÞii
¼hh
N
h
k
1
ð
u
Þ;wii þ hh
N
u
;h
k
2
ðwÞii
¼hh
N
h
k
1
ð
u
Þ;wii þ hhh
k
2
N
u
;wii
¼ hhð
N
h
k
1
þh
k
2
N
Þ
u
;wii
which completes the proof. h
Let
U
2F

h
i
ðN
0
i
Þ;i¼1;2. Then the multiplication opera-
tor by
U
(see Ref. [15]) is defined by
hhM
U
f;gii ¼ hh
U
;f:gii;f;g2F
h
i
ðN
0
i
Þ:
We observe that for
U
ð
U
n
Þ
nP0
and /
0
ð1;0;0;Þ (i.e,
/
0
ðxÞ¼1), we have MU/
0
¼
U
. Moreover,
U
#MUyields
a continuous injection from F

h
i
ðN
0
i
Þinto LðF
h
i
ðN
0
i
Þ;
F

h
i
ðN
0
i
ÞÞ and if
u
2F
h
i
ðN
0
i
Þthen M
u
belongs to LðF
h
i
ðN
0
i
Þ;
F
h
i
ðN
0
i
ÞÞ.
Theorem 3.3. Let
U
2F

h
i
ðN
0
i
Þ;i¼1;2, then
ðe
H
k
ðM
U
ÞÞ/
0
¼h
k
ð
U
Þ
for all k2ð0;1Þ.
Proof. Recall from [21] that
Q
s;t
¼q
s
q
t
;
8
s;tP0
in particular,
Q
t
¼q
t
q
t
;
8
tP0:
H. Rguigui / Chaos, Solitons & Fractals 73 (2015) 80–89 85

Which gives
hh e
H
k
ð
N
Þ/
0
;wii ¼ hhH
k
Kð
N
Þ;/
0
wii
¼Z
1
0
e
kt
hhðq
t
q
t
ÞKð
N
Þ;/
0
wiidt
¼Z
1
0
e
kt
hhKð
N
Þ;ðq
t
/
0
Þðq
t
wÞiidt
¼Z
1
0
e
kt
hh
N
ðq
t
/
0
Þ;q
t
wiidt
But ðq
t
/
0
Þ¼/
0
, then
hh e
H
k
ð
N
Þ/
0
;wii ¼ Z
1
0
e
kt
hh
N
/
0
;q
t
wiidt
¼Z
1
0
e
kt
hhq
t
N
/
0
;wiidt ¼hhh
k
N
/
0
;wii:
Replacing
N
by MU, we obtain
e
H
k
ðM
U
Þ

/
0
¼h
k
M
U
/
0
¼h
k
ð
U
Þ:
This gives the desired statement. h
4. Integral representations
Recall from ([2,18]) that a test function
u
2F
h
1
;h
2
ðN
0
1
N
0
2
Þis positive if it satisfies the following condition
u
ðx;yÞP0;
8
ðx;yÞ2X
0
1
X
0
2
:
Let F
h
1
;h
2
ðN
0
1
N
0
2
Þ
þ
denote the following set
F
h
1
;h
2
ðN
0
1
N
0
2
Þ
þ
:¼
u
2F
h
1
;h
2
ðN
0
1
N
0
2
Þ;
u
ðx;yÞP0;
8
ðx;yÞ2X
0
1
X
0
2

:
Note that under condition (2), we have the following Gelf-
and triplet
F
h
1
;h
2
ðN
0
1
N
0
2
ÞL
2
ðX
0
1
X
0
2
;
l
1

l
2
ÞF

h
1
;h
2
ðN
0
1
N
0
2
Þ
for more details see [8]. A generalised function
U
2F

h
1
;h
2
ðN
0
1
N
0
2
Þis said to be positive in the usual sense,
if it satisfies the following condition
hh
U
;
u
ii P0;
8
u
2F
h
1
;h
2
ðN
0
1
N
0
2
Þ
þ
:
Let
u
;w2F
h
1
;h
2
ðN
0
1
N
0
2
Þ
þ
. Then, we have
hhw;
u
ii ¼ Z
X
0
1
X
0
2
wðx;yÞ
u
ðx;yÞd
l
1
ðxÞd
l
2
ðyÞP0
because
u
ðx;yÞand wðx;yÞare positive for all ðx;yÞ2
X
0
1
X
0
2
. This shows that any positive test function wis also
a positive generalised function (as should be). An operator
N
2 LðF

h
1
ðN
0
1
Þ;F
h
2
ðN
0
2
ÞÞ is positive if its kernel Kð
N
Þis an
element of F
h
1
;h
2
ðN
0
1
N
0
2
Þ
þ
. We denote by
LðF

h
1
ðN
0
1
Þ;F
h
2
ðN
0
2
ÞÞ
þ
:
¼
N
2 LðF

h
1
ðN
0
1
Þ;F
h
2
ðN
0
2
ÞÞ=Kð
N
Þ2F
h
1
;h
2
ðN
0
1
N
0
2
Þ
þ
no
:
Theorem 4.1. Let
N
2 LðF

h
1
ðN
0
1
Þ;F
h
2
ðN
0
2
ÞÞ
þ
, then the solu-
tions
P
t
,
K
s
and
!
t
(of the Cauchy problems ((14), (15)and
(16), respectively)) are positive. In this case,
P
t
,
K
s
and
!
t
have the following representations
hhh
P
t
;Tiii ¼ Z
X
0
1
X
0
2
KðTÞðx;yÞd
l
e
Q
t
ðNÞ
ðx;yÞ;ð37Þ
hhh
K
s
;Tiii ¼ Z
X
0
1
X
0
2
KðTÞðx;yÞd
l
e
Q
s;0
ðNÞ
ðx;yÞ;ð38Þ
hhh!
t
;Tiii ¼ Z
X
0
1
X
0
2
KðTÞðx;yÞd
l
e
Q
0;t
ðNÞ
ðx;yÞ;ð39Þ
respectively, for all T 2 LðF

h
1
ðN
0
1
Þ;F
h
2
ðN
0
2
ÞÞ, where
l
e
Q
t
ð
N
Þ
;
l
e
Q
0;s
ð
N
Þ
and
l
e
Q
0;t
ð
N
Þ
are unique positive Borel measures on
X
0
1
X
0
2
such that its laplace transforms are given by
Lð
l
e
Q
t
ðNÞ
Þðn;
g
Þ ¼ hhh
P
t
;
N
g
;n
s
iii ¼ Kð
P
t
Þðn;
g
Þ;n2N
1
;
g
2N
2
Lð
l
e
Q
s;0
ðNÞ
Þðn;
g
Þ ¼ hhh
K
s
;
N
g
;n
s
iii ¼ Kð
K
s
Þðn;
g
Þ;n2N
1
;
g
2N
2
Lð
l
e
Q
0;t
ðNÞ
Þðn;
g
Þ ¼ hhh!
t
;
N
g
;n
s
iii ¼ Kð!
t
Þðn;
g
Þ;n2N
1
;
g
2N
2
respectively, and Kð
N
g
;n
s
Þ¼e
n
e
g
.
Proof. Applying the kernel to the solution
P
t
, we get
Kð
P
t
Þðx;yÞ¼Q
t
Kð
N
Þðx;yÞ¼wðe
t
x;e
t
yÞ
for all ðx;yÞ2X
0
1
X
0
2
, where w¼Kð
N
Þ. Then, by definition,
P
t
is positive iff Kð
P
t
Þis positive iff Q
t
Kð
N
Þis positive.
Then, if
N
is positive, we have
P
t
is positive. On the other
hand, using Theorem 3.2 in ([2]) and in view of (33),we
obtain the representation (37), where
Lð
l
e
Q
t
ðNÞ
Þðn;
g
Þ ¼ hhKð
N
t
Þ;e
n
e
g
ii
¼ hhh
P
t
;K
1
ðe
n
e
g
Þiii;n2N
1
;
g
2N
2
:
Similarly, we complete the proof. h
Theorem 4.2. Let
N
2 LðF

h
1
ðN
0
1
Þ;F
h
2
ðN
0
2
ÞÞ
þ
. Then, the oper-
ators e
H
k
ð
N
Þand e
H
g
k
1
;k
2
ð
N
Þare positive. Moreover, it have the
following integral representations
hhh e
H
k
ð
N
Þ;Tiii ¼ Z
1
0
Z
X
0
1
X
0
2
e
kt
KðTÞðx;yÞd
l
e
Q
t
ðNÞ
ðx;yÞdt;
hhh e
H
g
k
1
;k
2
ð
N
Þ;Tiii ¼ Z
1
0
Z
X
0
1
X
0
2
e
k
1
s
KðTÞðx;yÞd
l
e
Q
s;0
ðNÞ
ðx;yÞds
þZ
1
0
Z
X
0
1
X
0
2
e
k
2
t
KðTÞðx;yÞd
l
e
Q
0;t
ðNÞ
ðx;yÞdt;
respectively, for all T 2 LðF

h
1
ðN
0
1
Þ;F
h
2
ðN
0
2
ÞÞ, where
l
e
Q
t
ð
N
Þ
;
l
e
Q
0;s
ð
N
Þ
and
l
e
Q
0;t
ð
N
Þ
are unique positive Borel measures on
X
0
1
X
0
2
such that its laplace transforms are given by
L
l
e
Q
t
ðNÞ

ðn;
g
Þ ¼ hhh
P
t
;
N
g
;n
s
iii ¼ Kð
P
t
Þðn;
g
Þ;n2N
1
;
g
2N
2
L
l
e
Q
s;0
ðNÞ

ðn;
g
Þ ¼ hhh
K
s
;
N
g
;n
s
iii ¼ Kð
K
s
Þðn;
g
Þ;n2N
1
;
g
2N
2
L
l
e
Q
0;t
ðNÞ

ðn;
g
Þ ¼ hhh!
t
;
N
g
;n
s
iii ¼ Kð!
t
Þðn;
g
Þ;n2N
1
;
g
2N
2
86 H. Rguigui / Chaos, Solitons & Fractals 73 (2015) 80–89

respectively, Kð
N
g
;n
s
Þ¼e
n
e
g
,
P
t
,
K
s
and
!
t
are the solutions
of the Cauchy problems (14), (15)and (16), respectively.
Proof. Using Theorem 4.1 we can show the positivity of
e
H
k
ð
N
Þ. Moreover, we have
hhh e
H
k
ð
N
Þ;Tiii ¼ hhH
k
Kð
N
Þ;KðTÞii
¼Z
1
0
e
kt
hhQ
t
Kð
N
Þ;KðTÞiidt
¼Z
1
0
e
kt
hhh
P
t
;Tiiidt
¼Z
1
0
Z
X
0
1
X
0
2
e
kt
KðTÞðx;yÞd
l
e
Q
t
ðNÞ
ðx;yÞdt:
Similarly, we complete the proof. h
Similarly to Theorem 4.2, we get the following.
Theorem 4.3. Let
N
2 LðF

h
1
ðN
0
1
Þ;F
h
2
ðN
0
2
ÞÞ
þ
. Then, the oper-
ators e
Gð
N
Þand e
G
g
ð
N
Þare positive. Moreover, we have
hhhe
Gð
N
Þ;Tiii ¼ Z
1
0
Z
X
0
1
X
0
2
KðTÞðx;yÞd
l
e
Q
t
ðNN
0;0
Þ
ðx;yÞdt
hhhe
G
g
ð
N
Þ;Tiii ¼ Z
1
0
Z
X
0
1
X
0
2
KðTÞðx;yÞd
l
e
Q
s;0
ðNN
0;0
N

0
Þ
ðx;yÞds
þZ
1
0
Z
X
0
1
X
0
2
KðTÞðx;yÞd
l
e
Q
0;t
ðNN
0;0
N
þ
0
Þ
ðx;yÞdt
for all T 2 LðF

h
1
ðN
0
1
Þ;F
h
2
ðN
0
2
ÞÞ, where
l
e
Q
t
ð
N

N
0;0
Þ
;
l
e
Q
s;0
ð
N

N
0;0
Þ
and
l
e
Q
0;t
ð
N

N
0;0
Þ
are unique positive Borel measures on
X
0
1
X
0
2
such that its laplace transforms are given by
L
l
e
Q
t
ðNN
0;0
Þ

ðn;
g
Þ¼Q
t
ðKð
N

N
0;0
ÞÞðn;
g
Þ;n2N
1
;
g
2N
2
L
l
e
Q
s;0
ðNN
0;0
N

0
Þ

ðn;
g
Þ¼Q
s;0
ðKð
N

N
0;0

N

0
ÞÞðn;
g
Þ;
n2N
1
;
g
2N
2
L
l
e
Q
0;t
ðNN
0;0
N
þ
0
Þ

ðn;
g
Þ¼Q
0;t
ðKð
N

N
0;0

N
þ
0
ÞÞðn;
g
Þ;
n2N
1
;
g
2N
2
;
respectively.
5. Markovianity of the quantum potentials
Recall from [17] (see also [21]) that F
h
1
;h
2
ðN
0
1
N
0
2
Þis a
nuclear algebra with the involution * defined by
u

ðz;wÞ:¼
u
ðz;wÞ;z2N
0
1
;w2N
0
2
for all
u
2F
h
1
;h
2
ðN
0
1
N
0
2
Þ. Using the isomorphism K,we
can define the involution (denoted by the same symbol *)
on LðF

h
1
ðN
0
1
Þ;F
h
2
ðN
0
2
ÞÞ as follows
N

:¼K
1
ððKð
N
ÞÞ

Þ;
8
N
2 LðF

h
1
ðN
0
1
Þ;F
h
2
ðN
0
2
ÞÞ
Since F
h
1
;h
2
ðN
0
1
N
0
2
Þis closed under multiplication, there
exists a unique elements
u
2F
h
1
;h
2
ðN
0
1
N
0
2
Þsuch that
u
¼Kð
N
1
ÞKð
N
2
Þ
Then, by the topology isomorphism K, there exists
N
2 LðF

h
1
ðN
0
1
Þ;F
h
2
ðN
0
2
ÞÞ such that
Kð
N
Þ¼Kð
N
1
ÞKð
N
2
Þð40Þ
which is equivalent to
N
¼K
1
Kð
N
1
ÞKð
N
2
ÞðÞ ð41Þ
Denoted by
N
to be the product between
N
1
and
N
2
, i.e,
N
¼
N
1

N
2
Note that from (40) we see that the product is commu-
tative. Now, define the following cones
B:¼
N


N
;
N
2 LðF

h
1
ðN
0
1
Þ;F
h
2
N
0
2

Þ
no

Elements in B are said to be B-positive operators, see Ref.
[21].
Remark 3. Let
N
2B, then there exists T2 LðF

h
1
ðN
0
1
Þ;
F
h
2
ðN
0
2
ÞÞ such that
N
¼T

T. Applying the kernel map K,
we get
Kð
N
Þ ¼ ðKðTÞÞ

KðTÞ:
But, we know that
ðKðTÞÞ

¼ KðK
1
ððKðTÞÞ

ÞÞ ¼ ðKðTÞÞ

:
Then, we obtain
Kð
N
Þ ¼ ðKðTÞÞ

KðTÞ:
This leads to
Kð
N
Þðxþi0;yþi0Þ ¼ ðKðTÞÞ

ðxþi0;yþi0ÞKðTÞðxþi0;yþi0Þ
¼KðTÞðxþi0;yþi0ÞKðTÞðxþi0;yþi0Þ
¼KðTÞðxþi0;yþi0Þ
jj
2
P0
From which we deduce that Kð
N
Þ2F
h
1
;h
2
ðN
0
1
N
0
2
Þ
þ
, which
is equivalent to say that
N
2 LðF

h
1
ðN
0
1
Þ;F
h
2
ðN
0
2
ÞÞ
þ
. We con-
clude that, any B-positive operator
N
is also a positive
operator, i.e. B LðF

h
1
ðN
0
1
Þ;F
h
2
ðN
0
2
ÞÞ
þ
.
Let Bdefined by
B:¼
K
2 LðF

h
1
ðN
0
1
Þ;F
h
2
ðN
0
2
ÞÞ;hhh
K
;
N
iii P0;
8
N
2B
no
:
For S;T2 LðF

h
1
ðN
0
1
Þ;F
h
2
ðN
0
2
ÞÞ, we say that S6T,if
TS2B. Denoted by I
0
¼K
1
ð1
F
h1;h2
ðN
0
1
N
0
2
Þ
Þ. Motivated by
the definition of Markovian operator in [7] (see also
[21]), we can define the Markovianity as follows.
Definition 5.1. A map P: LðF

h
1
ðN
0
1
Þ;F
h
2
ðN
0
2
ÞÞ ! LðF

h
1
ðN
0
1
Þ;
F
h
2
ðN
0
2
ÞÞ is said to be
(i) positive if PðBÞ #B
(ii) Markovian if it is positive and Pð
N
Þ6I
0
whenever
N
¼
N

and
N
6I
0
.
H. Rguigui / Chaos, Solitons & Fractals 73 (2015) 80–89 87

Theorem 5.1. For k2½1;1Þ and k
1
;k
2
2ð0;1Þ such that
k
1
k
2
Pk
1
þk
2
, the quantum k-potential and the generalised
quantum ðk
1
;k
2
Þ-potential are Markovian.
Proof. Let
N
2B, then there exists S2 LðF

h
1
ðN
0
1
Þ;F
h
2
ðN
0
2
ÞÞ
such that
N
¼S

S
Then, for all s;tP0;z2N
0
1
and w2N
0
2
, we have
Kg
Q
s;t
ð
N
Þðz;wÞ¼Q
s;t
ðKðK
1
ðKðS

ÞKðSÞÞÞÞðz;wÞ
¼Q
s;t
ððKðS

ÞÞKðSÞÞðz;wÞ
¼ ðKðS

ÞKðSÞÞðe
s
z;e
t
wÞ
¼Q
s;t
ðKðS

ÞÞðz;wÞQ
s;t
ðKðSÞÞðz;wÞ:
Using (41), we get
g
Q
s;t
ð
N
Þ¼K
1
ðQ
s;t
ðKðS

ÞÞQ
s;t
ðKðSÞÞÞ
¼K
1
ðKðg
Q
s;t
ðS

ÞÞKð e
Q
s;t
ðSÞÞÞ
¼g
Q
s;t
ðS

Þg
Q
s;t
ðSÞ:
On the other hand, we have
Kg
Q
s;t
ðS

Þ

¼Q
s;t
ðKðS

ÞÞ
But we know that
S

¼K
1
ðKðSÞÞ

ðÞ
Then, we get
Kðg
Q
s;t
ðS

ÞÞðz;wÞ¼Q
s;t
ððKðSÞÞ

Þðz;wÞ
¼KðSÞðÞ

ðe
s
z;e
t
wÞ
¼KðSÞe
s
z;e
t
wðÞ
¼ðQ
s;t
KðSÞÞ z;wðÞ
¼ðQ
s;t
KðSÞÞ

ðz;wÞ:
From which we obtain
g
Q
s;t
ðS

Þ¼K
1
ðQ
s;t
KðSÞÞ


¼K
1
ðKg
Q
s;t
ðSÞÞ


¼g
Q
s;t
ðSÞ


:
ð42Þ
Hence, we get
g
Q
s;t
ð
N
Þ¼ g
Q
s;t
ðSÞ


g
Q
s;t
ðSÞ
This proves that g
Q
s;t
ð
N
Þ2Bfor all s;tP0. On the other
hand, let
K
2B. Then, using Remark 2, we get
e
H
g
k
1
;k
2
ð
K
Þ;
N
DEDEDE
¼
K
;e
H
g
k
1
;k
2
ð
N
Þ
DEDEDE
for all
N
2B. Then
hhh
K
;e
H
g
k
1
;k
2
ð
N
Þiii ¼ hhKð
K
Þ;H
g
k
1
;k
2
ðKð
N
ÞÞii
¼Z
1
0
e
k
1
s
hhKð
K
Þ;Q
s;0
ðKð
N
ÞÞiids
þZ
1
0
e
k
2
t
hhKð
K
Þ;Q
0;t
ðKð
N
ÞÞiidt
¼Z
1
0
e
k
1
s
hhh
K
;g
Q
s;0
ð
N
Þiiids
þZ
1
0
e
k
2
t
hhh
K
;g
Q
0;t
ð
N
Þiiidt:
Therefore using the fact that g
Q
s;0
ð
N
Þ;g
Q
0;t
ð
N
Þ2B, we get
K
;g
Q
s;0
ð
N
Þ
DEDEDE
P0and
K
;g
Q
0;t
ð
N
Þ
DEDEDE
P0:
From which we obtain
e
H
g
k
1
;k
2
ð
K
Þ;
N
DEDEDE
P0:
This gives e
H
g
k
1
;k
2
ð
K
Þ2B, which complete the positivity of
e
H
g
k
1
;k
2
.
Let
K
2 LðF

h
1
ðN
0
1
Þ;F
h
2
ðN
0
2
ÞÞ such that
K
6I
0
and
K
¼
K

. This gives I
0

K
2B. But we know that e
H
g
k
1
;k
2
is
positive operator. Then, e
H
g
k
1
;k
2
ð
K
Þ
2B. From which we get
e
H
g
k
1
;k
2
ðI
0

K
Þ;
N
DEDEDE
P0
8
N
2B:
This leads to
1
k
1
þ1
k
2

I
0
e
H
g
k
1
;k
2
ð
K
Þ;
N

P0
8
N
2B:ð43Þ
But we know that
1
k
1
þ
1
k
2

61, then
I
0
1
k
1
þ1
k
2

I
0
;
N

¼11
k
1
þ1
k
2

hhhI
0
;
N
iii
¼11
k
1
þ1
k
2

hh1;Kð
N
Þii
¼11
k
1
þ1
k
2

Z
X
0
1
X
0
2
Kð
N
Þðx;yÞd
l
1
ðxÞd
l
2
ðyÞ:
Using Remark 3, since
N
2Band 1 
1
k
1
þ
1
k
2

P0, we
obtain
I
0
1
k
1
þ1
k
2

I
0
;
N

P0
8
N
2B:ð44Þ
By a simple addition of (43) and (44), we get
I
0
e
H
g
k
1
;k
2
ð
K
Þ;
N
DEDEDE
P0
8
N
2B:
From which we deduce that I
0
e
H
g
k
1
;k
2
ð
K
Þ2B, i.e.,
e
H
g
k
1
;k
2
ð
K
Þ6I
0
. This completes the proof of the Markovianity
of e
H
g
k
1
;k
2
. Similarly we show the Markovianity of the others
potentials. h
References
[1] Accardi L, Ouerdiane H, Smolyanov OG. Lévy Laplacian acting on
operators. Russ J Math Phys 2003;10(4):359–80.
[2] Ayed W, Ouerdiane H. Positive operators and integral representation.
Taiwanese J Math 2007;11(5):1457–75.
[3] Barhoumi A, Kuo H-H, Ouerdiane H. Generalized Gross heat equation
with noises. Soochow J Math 2006;32(1):113–25.
[4] Barhoumi A, Ouerdiane H, Rguigui H. QWN-Euler operator and
associated Cauchy problem. Infin Dimens Anal Quantum Probab
Relat Top 2012;15(1):1250004 (20 pages).
[5] Barhoumi A, Ouerdiane H, Rguigui H. Stochastic heat equation on
algebra of generalized functions. Infin Dimens Anal Quantum Probab
Relat Top 2012;15(4):1250026 (18 pages).
[6] Chung DM, Ji UC. Transform on white noise functionals with their
application to Cauchy problems. Nagoya Math J 1997;147:1–23.
[7] Cipriani F. Dirichlet Forms on Noncommutative Spaces, Quantum
Potential Theory. Berlin Heidelberg: Springer-Verlag; 2008. 161–
276.
88 H. Rguigui / Chaos, Solitons & Fractals 73 (2015) 80–89

[8] Gannoun R, Hachaichi R, Ouerdiane H, Rezgui A. Un théorème de
dualité entre espace de fonctions holomorphes à croissance
exponentielle. J Funct Anal 2002;171(1):1–14.
[9] Hida T, Kuo H-H, Pathoff J, Streit L. White Noise. An Infinite
Dimensional Calculus. Dordrecht: Kluwer Academic Publishers;
1993.
[10] Ji UC. Quantum extensions of Fourier–Gauss and Fourier–Mehler
transforms. J Korean Math Soc 2008;45(6):1785–801.
[11] Ji UC, Obata N, Ouerdiane H. Analytic characterization of generalized
fock space operator as two-variable entire function with growth
condition. World Sci 2002;5(3):395–407.
[12] Kang SJ. Heat and Poisson equations associated with number
operator in white noise analysis. Soochow J Math 1994;20:
45–55.
[13] Kuo H-H. Potential theory associated with Uhlenbeck – Ornstein
process. J Funct Anal 1976;21:63–75.
[14] Kuo H-H. Withe Noise Distribution Theory. CRC Press; 1996.
[15] Obata N. White noise calculus and fock space. Lect Notes Math.
1994;1577.
[16] N. Obata, Quantum white noise calculus based on nuclear algebras of
entire function, Trends in Infinite Dimensional Analysis and
Quantum Probability (Kyoto 2001), RIMS No. 1278, 130-157.
[17] Ouerdiane H. Infinite dimensional entire functions and application
to stochastic differential equations. Not South Afr Math Soc 2004;
35(1):23–45.
[18] Ouerdiane H, Rezgui A. Representation integrale de fonctionnelles
analytiques positives. Can Math Soc Conf Proc 2000;28:283–90.
[19] Ouerdiane H, Rguigui H. QWN-conservation operator and associated
Wick differential equation. Commun Stoch Anal 2012;6(3):437–50.
[20] Piech MA. Parabolic equations associated with the number operator.
Trans Amer Math Soc 1974;194:213–22.
[21] Rguigui H. Quantum Ornstein–Uhlenbeck semigroups, Quantum
Studies: Mathematics and Foundations; 2014. http://dx.doi.org/10.
1007/s40509-014-0023-5.
H. Rguigui / Chaos, Solitons & Fractals 73 (2015) 80–89 89